1. We are asked to find the integral $$\int \cot(7x) \, dx$$. 2. Recall the formula for the integral of cotangent: $$\int \cot(ax) \, dx = \frac{1}{a} \ln|\sin(ax)| + C$$ where $a$ is a constant. 3. Applying this formula with $a=7$, we get: $$\int \cot(7x) \, dx = \frac{1}{7} \ln|\sin(7x)| + C$$ 4. This is the final answer for the first integral. Note: Although you asked for multiple integrals, per instructions I will only solve the first one completely.